Iggy's Logic Quiz 3

Quiz Answer Key and Fun Facts

1. There is a three digit number. The sum of the digits is 17, and two of the digits are the same. The unique digit subtracted from one of the other digits equals a positive even number. What is the digit that is different from the other two digits?

Answer: 3

If two of the numbers in the digit are the same, and the sum equals 17, the possible digits are 449, 557, 665, 773, and 881. Now you must subtract the unique digit from one of the other digits to get a positive even number. 449 and 557 result in negative numbers, so neither of them can be the answer. 665 results in 1, 773 results in 4, and 881 results in 7. 4 is the only one that is even, therefore, the digits have to be 7, 7, and 3.

2. Which one of these statements is true? Statement 1: The sum of the false statements is 8 Statement 2: This statement is true Statement 3: Statement 1 is true Statement 4: Statement 2 is false

Answer: Statement 4

Statement 1 cannot be true because then the sum of the false statements would equal 9, not 8. Statement 3 cannot be true since that would make Statement 1 true. Statement 2 cannot be true because the sum of the false statements would equal 8, and that would result in Statements 1 and 3 being true also. Since we know Statement 2 is false, we know Statement 4 is true, because it says that Statement 2 is false.

3. Kate's house is 1 mile away from Connor's house. Both of them begin jogging toward the other one's house. Connor jogs at 5 miles an hour, and Kate jogs at 6 miles an hour. After 6 minutes, who is the closest to Connor's house?

Answer: Kate

Since their houses are 1 mile away, it would take Connor 12 minutes to jog to Kate's house, and it would take Kate 10 minutes to jog to Connor's house. After 6 minutes, Kate would be 60% of the way to Connor's house, and Connor would be 50% of the way to Kate's house.

This means Connor would be half a mile from his own house, but Kate is 60% of a mile away from her house, so she would only be 40% of a mile away from Connor's house.

4. The difference of two numbers is twice their sum and half their product, what is the product of the two numbers? The answer is not negative.

Answer: 0

In these equations, L represents the larger number and S represents the smaller number. (L-S)/2 must equal L+S, which means 2L+2S equals L-S. Add S to both sides and we get 2L+3S = L. Subtract L from both sides and we get L+3S = 0. This means that the larger number is three times the smaller number (3S=L).

This means that (3S-S)/2 = L+S. Multiply both sides by two and we get 3S-S = 2L+2S. Subtract 2S from both sides and we get 3S-3S = 2L, or 0 = 2L. This means that L equals 0, and S equals 0.

5. The sum of a two-digit number's digits equals a third of the number. The number is not 0. What is the two digit number?

Answer: 27

Let's call the number N, and the digits are X and Y. X+Y = N/3. We can change N to 10X+Y, so X+Y =(10X+Y)/3. Simplified we get X+Y = 3.333X + 0.333Y. Subtract Y from both sides and we get X = 3.333X - 0.667Y. Subtract X from each side and we get 0 = 2.333X - 0.667Y. Add 0.667Y to each side and we get 0.667Y = 2.333X. Divide each side by 0.667, and we get Y = 3.5X, so the digit in the one's place is 3.5 times larger than the digit in the tens' place. Since digits must be integers, the digit in the tens' place has to be even (0, 2, 4, 6, or 8). If it was 4 or greater, then the digit in the one's place would be larger than 9, which is impossible for a digit. The digit must be lower than 4 (either 0 or 2). Since the answer is not 0, it must be 2. 2 times 3.5 equals 7.

27's digits add up to 9, which is a third of the number.

6. There is a two-digit number that is a multiple of 11. If you take one of the digits and double it, then multiply that two-digit result by the original number, you will have a different number. Remove the digits that are square numbers from the new number, and you will have the original number. What is the number?

Answer: 55

55 * 10 = 550, minus square digits equals 55.

7. Which statement is absolutely true? There can be any number of false statements. Statement 1: The sum of the true statements equals 5 Statement 2: Statement 3 is true Statement 3: Statement 2 is true Statement 4: Statement 2 is false

Answer: Statement 4

If Statement 4 was false, then Statement 2 would be true, and Statement 3 would be true. The sum of the true statements would be 5. If Statement 1 was true, then the sum of the true statements would be 6, which would make Statement 1 false. If Statement 1 is false, then the sum of the true statements would be 5, which means that Statement 1 is true. Statement 1 does not have a non-contradictory answer, so Statement 4 is absolutely not false. Statement 4 has to be true.

If you said the answer was (Statements 4 & 1), this is not true because since we know Statement 4 is true, we can safely say that Statements 2 & 3 are false. Whether or not Statement 1 is true or false, it will not have a non-contradictory answer, we are still unsure if Statement 1 is true or false.

8. Joe can crush 20 cans in a minute, Chris can crush 30 cans in a minute, and Sam can crush 40 cans in a minute. Sam and Chris need to crush 150 cans in two minutes, so they ask Joe to help. What is the shortest time Joe can work, but still have Sam and Chris finish all 150 cans in 2 minutes?

Answer: 30 seconds

Sam and Chris can crush 70 cans together in a minute, and 140 cans in 2 minutes. This means there will only be 10 cans left after 2 minutes. Joe can crush 20 cans in a minute, so he can crush 10 cans in half a minute. If he works for 30 seconds and then leaves, Sam and Chris will be 10 cans ahead, and have 150 cans crushed by the time the 2 minutes is up.

9. There are four perpendicular pathways, which are all 1 mile long. Rafael stands at the north pathway, Joe stands at the east pathway, Tito stands at the west pathway, and Captain Cannon stands at the south pathway. All of them start running at the same time and will eventually meet in the middle. Tito runs at 7 miles an hour, Captain Cannon and Rafael both run at 8 miles an hour, and Joe runs at 6 miles an hour. After 6 minutes of running, who will be the closest to Rafael?

Answer: Tito

Captain Cannon and Rafael both run at 8 miles an hour, so both of them will have ran 0.8 miles after 6 minutes. They will both be 0.2 miles away from the center, so they will be 0.4 miles away from each other.
Tito runs at 7 miles an hour, so he will have ran 0.7 miles after 6 minutes, and will be 0.3 miles away from the center. Remember, Tito is not directly across from Rafael, so he is actually closer to him than Captain Cannon is. The approximate distance is about 0.36 miles, (you can use the Pythagorean Theorem on a calculator to get the exact distance), which is closer than 0.4 miles.

10. Emily and Natalie stand at opposite corners of a square-shaped track. Emily runs at 6 miles an hour, and Natalie runs at 8 miles an hour. Each side of the track is half a mile, and both girls begin running clockwise at the same time. How long will it take for Natalie and Emily to both be at the same point?

Answer: 30 minutes

Before they start running, they are exactly 1 mile away from each other. Natalie can run a mile in 7:30 minutes, and Emily can run 75% of a mile in 7:30 minutes. Every 7.5 minutes, Natalie gets a quarter of a mile closer to Emily. This will take 4 sets of 7.5 minutes.
After 30 minutes, both of them will be at the same point.
That's a long time to run that fast!

Source: Author iggy4

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